A linesearch projection algorithm for solving equilibrium problems without monotonicity in Hilbert spaces. (English) Zbl 1524.65239
Summary: We propose a linesearch projection algorithm for solving non-monotone and non-Lipschitzian equilibrium problems in Hilbert spaces. It is proved that the sequence generated by the proposed algorithm converges strongly to a solution of the equilibrium problem under the assumption that the solution set of the associated Minty equilibrium problem is nonempty. Compared with existing methods, we do not employ Fejér monotonicity in the strategy of proving the convergence. This comes from projecting a fixed point instead of the current point onto a subset of the feasible set at each iteration. Moreover, employing an Armijo-linesearch without subgradient has a great advantage in CPU-time. Some numerical experiments demonstrate the efficiency and strength of the presented algorithm.
MSC:
| 65K15 | Numerical methods for variational inequalities and related problems |
| 90C33 | Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming) |
| 65J15 | Numerical solutions to equations with nonlinear operators |
Keywords:
nonmonotone equilibrium problem; minty equilibrium problem; projection algorithm; Armijo-linesearch; strong convergenceReferences:
| [1] | P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, 62, 271-283 (2013) · Zbl 1290.90084 · doi:10.1080/02331934.2011.607497 |
| [2] | P. N. Anh, A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems, Bull. Malays. Math. Sci. Soc., 36, 107-116 (2013) · Zbl 1263.65066 |
| [3] | P. N. L. T. H. Anh An, The subgradient extragradient method extended to equilibrium problems, Optimization, 64, 225-248 (2015) · Zbl 1317.65149 · doi:10.1080/02331934.2012.745528 |
| [4] | H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, New York, 2011. · Zbl 1218.47001 |
| [5] | J. Y. R. H. M. Bello-Cruz Díaz Millán Phan, Conditional extragradient algorithms for solving variational inequalities, Pacific J. Optim., 15, 331-357 (2019) · Zbl 1458.49011 |
| [6] | G. M. M. M. Bigi Castellani Pappalardo Passacantando, Existence and solution methods for equilibria, Eur. J. Oper. Res., 227, 1-11 (2013) · Zbl 1292.90315 · doi:10.1016/j.ejor.2012.11.037 |
| [7] | G. M. Bigi Passacantando, Auxiliary problem principles for equilibria, Optimization, 66, 1955-1972 (2017) · Zbl 1410.90212 · doi:10.1080/02331934.2016.1227808 |
| [8] | E. W. Blum Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63, 123-145 (1994) · Zbl 0888.49007 |
| [9] | R. S. R. Burachik Díaz Millán, A projection algorithm for non-monotone variational inequalities, Set-Valued Var. Anal., 28, 149-166 (2020) · Zbl 1434.90201 · doi:10.1007/s11228-019-00517-0 |
| [10] | Y. A. S. Censor Gibali Reich, Strong convergence of subgradient extragradient methods for variational inequality problem in Hilbert space, Optim. Methods Softw., 26, 827-845 (2011) · Zbl 1232.58008 · doi:10.1080/10556788.2010.551536 |
| [11] | L. M. R. Y. P. Deng Hu Fang, Projection extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces, Numer. Algor., 86, 191-221 (2021) · Zbl 1456.65036 · doi:10.1007/s11075-020-00885-x |
| [12] | B. V. L. D. Dinh Muu, A projection algorithm for solving pseudomonotone equilibrium problems and it’s application to a class of bilevel equilibria, Optimization, 64, 559-575 (2015) · Zbl 1317.65152 · doi:10.1080/02331934.2013.773329 |
| [13] | B. V. D. S. Dinh Kim, Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space, J. Comput. Appl. Math., 302, 106-117 (2016) · Zbl 1334.90125 · doi:10.1016/j.cam.2016.01.054 |
| [14] | K. Fan, A minimax inequality and applications, Inequalities, Academic Press, New York, 3, 103-113 (1972) · Zbl 0302.49019 |
| [15] | S. D. A. S. Flam Antipin, Equilibrium programming using proximal-like algorithms, Math. Program., 78, 29-41 (1997) · Zbl 0890.90150 · doi:10.1016/S0025-5610(96)00071-8 |
| [16] | N. S. Hadjisavvas Schaible, Quasimonotone variational inequalities in Banach spaces, J. Optim. Theory Appl., 90, 95-111 (1996) · Zbl 0904.49005 · doi:10.1007/BF02192248 |
| [17] | D. V. Hieu, P. K. Quy, L. T. Hong and L. V. Vy, Accelerated hybrid methods for solving pseudomonotone equilibrium problems, Adv. Comput. Math., 46 (2020), Paper No. 58, 24 pp. · Zbl 1455.65084 |
| [18] | A. N. W. Iusem Sosa, New existence results for equilibrium problems, Nonlinear Anal. TMA., 52, 621-635 (2003) · Zbl 1017.49008 · doi:10.1016/S0362-546X(02)00154-2 |
| [19] | A. N. W. Iusem Sosa, On the proximal point method for equilibrium problem in Hilbert spaces, Optimization, 59, 1259-1274 (2010) · Zbl 1206.90212 · doi:10.1080/02331931003603133 |
| [20] | I. V. Konnov, Application of the proximal point method to nonmonotone equilibrium problems, J. Optim. Theory Appl., 119, 317-333 (2003) · Zbl 1084.49009 · doi:10.1023/B:JOTA.0000005448.12716.24 |
| [21] | G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Ekon. Mat. Metody., 12, 747-756 (1976) · Zbl 0342.90044 |
| [22] | J.-L. G. Lions Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20, 493-519 (1967) · Zbl 0152.34601 · doi:10.1002/cpa.3160200302 |
| [23] | C. H.-K. Martinez-Yanes Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. TMA., 64, 2400-2411 (2006) · Zbl 1105.47060 · doi:10.1016/j.na.2005.08.018 |
| [24] | G. Mastroeni, Gap functions for equilibrium problems, J. Glob. Optim., 27, 411-426 (2003) · Zbl 1061.90112 · doi:10.1023/A:1026050425030 |
| [25] | G. Mastroeni, On auxiliary principle for equilibrium problems, Equilibrium Problems and Variational Models, Nonconvex Optim. Appl., Kluwer Acad. Publ., Norwell, MA, 68, 289-298 (2003) · Zbl 1069.49009 · doi:10.1007/978-1-4613-0239-1_15 |
| [26] | A. Moudafi, On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces, J. Math. Anal. Appl., 359, 508-513 (2009) · Zbl 1176.90644 · doi:10.1016/j.jmaa.2009.06.005 |
| [27] | L. D. W. Muu Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal. TMA., 18, 1159-1166 (1992) · Zbl 0773.90092 · doi:10.1016/0362-546X(92)90159-C |
| [28] | L. D. T. D. Muu Quoc, Regularization algorithms for solving monotone Ky Fan inequalities with application to a Nash-Cournot equilibrium model, J. Optim. Theory Appl., 142, 185-204 (2009) · Zbl 1191.90084 · doi:10.1007/s10957-009-9529-0 |
| [29] | K. K. W. Nakajo Shimoji Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings in Banach spaces, J. Nonlinear Convex Anal., 8, 11-34 (2007) · Zbl 1125.49024 |
| [30] | T. D. P. N. L. D. Quoc Anh Muu, Dual extragradient algorithms extended to equilibrium problems, J. Glob. Optim., 52, 139-159 (2012) · Zbl 1258.90088 · doi:10.1007/s10898-011-9693-2 |
| [31] | T. D. L. D. Quoc Muu, Iterative methods for solving monotone equilibrium problems via dual gap functions, Comput. Optim. Appl., 51, 709-728 (2012) · Zbl 1268.90106 · doi:10.1007/s10589-010-9360-4 |
| [32] | R. T. Rockafellar, Convex Analysis, Princeton Mathematical Series, No. 28 Princeton University Press, Princeton, N.J., 1970 · Zbl 0193.18401 |
| [33] | P. S. Santos Scheimberg, An inexact subgradient algorithm for equilibrium problems, Comput. Appl. Math., 30, 91-107 (2011) · Zbl 1242.90265 |
| [34] | S. P. S. M. Scheimberg Santos, A relaxed projectionmethod for finite dimensional equilibrium problems, Optimization, 60, 1193-1208 (2011) · Zbl 1245.49041 · doi:10.1080/02331934.2010.527974 |
| [35] | J. J. P. T. T. T. V. Strodiot Vuong Nguyen, A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces, J. Glob. Optim., 64, 159-178 (2016) · Zbl 1357.90160 · doi:10.1007/s10898-015-0365-5 |
| [36] | D. Q. M. L. V. H. Tran Dung Nguyen, Extragradient algorithms extended to equilibrium problems, Optimization, 57, 749-776 (2008) · Zbl 1152.90564 · doi:10.1080/02331930601122876 |
| [37] | F. Q. N. J. Xia Huang, A projection-proximal point algorithm for solving generalized variational inequalities, J. Optim Theory Appl., 150, 98-117 (2011) · Zbl 1242.90267 · doi:10.1007/s10957-011-9825-3 |
| [38] | M. Y. Ye He, A double projection method for solving variational inequalities without monotonicity, Comput. Optim. Appl., 60, 141-150 (2015) · Zbl 1308.90184 · doi:10.1007/s10589-014-9659-7 |
| [39] | L. C. J. Y. Zeng Yao, Modified combined relaxation method for general monotone equilibrium problems in Hilbert spaces, J. Optim. Theory Appl., 131, 469-483 (2006) · Zbl 1139.90031 · doi:10.1007/s10957-006-9162-0 |
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